Preheat a grill or grill pan to medium high. Slice the chicken breasts 1/2 inch thick and toss with 2 tablespoons peanut sauce, 1/2 tablespoon vegetable oil, 1/4 teaspoon salt, and pepper to taste in a bowl. Grill the chicken until just cooked through, about 2 minutes per side. Transfer to a plate.
Whisk the remaining 5 tablespoons peanut sauce and 2 1/2 tablespoons vegetable oil, the lime juice and 2 tablespoons water in a large bowl. Add the lettuce, cucumber, carrot, bell pepper and cilantro and toss. Season with salt and pepper and divide among bowls. Top with the chicken, peanuts and more cilantro.
Per serving: Calories 363; Fat 20 g (Saturated 2 g); Cholesterol 66 mg; Sodium 208 mg; Carbohydrate 15 g; Fiber 5 g; Protein 32 g